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Free models of algebraic theories are always projective objects in the category of models, but the converse is not always true. For instance, some (actually, all) projective modules are direct summands of free modules, and these need not be free. On the other hand, the two notions do coincide for abelian groups.

Hence I'm left wondering: for which kinds of algebraic theories do free and projective coincide?

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  • $\begingroup$ It should be true that the projective objects are precisely the retracts of the free objects, so the question is equivalent to asking when retracts of free objects are free. This is apparently quite subtle: for example it's apparently an open question whether retracts of polynomial rings over fields are polynomial rings. See mathoverflow.net/questions/219938/…. $\endgroup$
    – Qiaochu Yuan
    Oct 17, 2015 at 22:34
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    $\begingroup$ @QiaochuYuan No longer a open question. The 2012 work of Gupta mentioned in the MO thread you link to gives a counterexample. Though as far as I know it's still open over fields of characteristic zero. $\endgroup$
    – Jeremy Rickard
    Oct 18, 2015 at 10:37

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